If we want to show that \(M\)  is \(P^*\) -convex, this amounts to finding, for each compact set \(K\) , a larger compact set \(K^{\prime }\)  such that the components of \(M\setminus K^{\prime }\)  are all unbounded, that is, having noncompact closure. This is because the boundary of the support of \(u\)  cannot intersect the region where \(P^*u=0\) , so the support of \(u\)  cannot extend into any unbounded component of \(M\setminus K^{\prime }\) . Thus the key topological lemma in this regard is the following one. The proof is thanks to Pierre PC [1]} and Aitor Iribar Lopez [2]} at mathoverflow.net. It is based on the proof of theorem 23.5 in [3]}.
